The generation and recording of borehole acoustic waves is a key measurement employed in oilfield wireline logging. Many borehole tools and methods are currently available for taking acoustic measurements. Some tools include a single source of sonic waves and two or more receivers; however, most of the tools now include many receivers arranged in an array. While the currently available acoustic tools are useful in providing a large range of information regarding the adjacent formation and the borehole parameters, a primary use of acoustic borehole measurements is the estimation of compressional wave formation slowness.
Compressional wave formation slowness is typically estimated using travel times acquired via a first motion detection process. In the case of a single source, two receiver tool suggested by the prior art, formation slowness is estimated by subtracting the arrival times between two receivers and dividing by the inter-receiver spacing. This estimate, however, is subject to inaccuracies due to tool tilt, borehole washouts, bed boundary effects, etc. Additional acoustic sources and receivers and more robust methods such as STC (Slowness-Time-Coherency analysis) among others have been used to reduce the inaccuracies introduced by such environmental effects.
The above described travel time measurement technique for determining formation slowness suffers from other shortfalls as well. Existing methods provide only one-dimensional values of formation slowness along the borehole axis and discards valuable information inherent in the signal regarding properties of the formation in other directions such as radial and/or azimuthal directions that are perpendicular to the borehole axis.
To overcome the difficulty in assessing the slowness distribution in both the axial and radial directions, a travel time tomographic inversion (namely “tomography”) can be employed. The aim of a travel time tomographic inversion is to find a solution or a model (i.e. formation slowness distribution along and around a borehole) that minimizes the discrepancy between measured travel times and calculated ones at all source-receiver pairs. The governing equation of travel time tomography is generally non-linear but can be solved by an iterative solution algorithm starting with an initial model. At each iteration of this solution process, the following two steps are executed: forward modeling and inversion. By “forward modeling”, the travel time is calculated for each source and receiver pair with a given model (from either an initial guess or the result of the previous iteration). However, since the model used is unlikely to be the true subsurface model, the calculated travel times are typically not consistent with the actual measured travel times. Hence, followed by this “forward modeling” step, the discrepancy between the actual measured and the calculated travel times is calculated and then minimized by an optimization/minimization method (e.g. a back-projection or conjugate-gradient) to update/adjust the previous resultant model. These two steps are conducted iteratively until a best fit is achieved (i.e. the travel time errors converge). Thus, the final solution model is obtained. A successful tomography depends strongly on these forward-modeling and inversion steps. As to each of these two steps, the numerical method used for the forward modeling is more critical and essential. It can be easily understood that an inaccurate and less robust modeling will generate incorrect travel times hence result in a wrong final solution.
U.S. Pat. No. 5,081,611 (B. Hornby, 1992) proposed such a tomography method based on inversion of travel times measured by a sonic tool. Hornby alleges that his method is able to determine slowness distributions away from the borehole. The method disclosed by Hornby, however, is not in frequent use, presumably because the modeling kernel based on a ray tracing technique for refraction waves is unreliable. In addition, the Hornby method requires implicitly imposed virtual layers along the borehole axis, which lacks robustness regarding prior information and geological support. Furthermore, the method taught by Hornby is limited to only two dimensions, i.e. along borehole axis and one of the radial directions.
As described previously, the modeling is a required step in this tomographic inversion and its robustness and efficiency is crucial to the success of the inversion. Although ray tracing is a well know numerical simulation technique and widely used in acoustic domain for travel time computations, there are numerous limitations of this method preventing its robust use for sonic tomography. For instance, ray tracing techniques assume the frequency of the sonic waves is infinitely high. Actual waves, however, frequently are band-limited. The propagation of actual waves is affected not only by the structures along the ray path as the ray approximation implies, but also by media in the vicinity of the ray path. For sonic waves, where the wavelength is not vastly smaller than the distance between a source and receivers, such a ray approximation is not accurate enough. Further, ray tracing computation is expensive, unstable and practically difficult to apply to a three-dimensional (3-D) tomography.
The present invention is directed to overcoming, or at least reducing the effects of, one or more of the problems outlined above.